Twin Composites, Strange Continued Fractions, and a Transformation that Euler Missed (Twice)

We introduce a polynomial E ( d , t , x ) in three variables that comes from the intersections of a family of ellipses described by Euler. For fixed odd integers t ≥ 3 , the sequence of E ( d , t , x ) with d running through the integers produces, conjecturally, sequences of “twin composites” analogous to the twin primes of the integers. This polynomial and its lower degree relative R ( d , t , x ) have strikingly simple discriminants and resolvents. Moreover, the roots of R for certain values of d have continued fractions with at least two large partial quotients, the second of which mysteriously involves the 12th cyclotomic polynomial. Various related polynomials whose roots also have conjecturally strange continued fractions are also examined.